I have been trying to understand this statement from Axler's Linear Algebra Done Right:
If $T:V \to W$ is a linear map with $v_1,...,v_n$ a basis of V and $w_1,...,w_n$ a basis of W, defined by $T(a_1v_1+...+a_nv_n)=a_1w_1+...+a_nw_n$, why is $T$ injective? Is it because $w_1,...,w_n$ is linearly independent?
What has the fact $w_1,...,w_n$ being linearly independent have to imply that T is injective? Is it because the nullspace is trivial.